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Subject: NECNON - the promised sequel rss

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Richard Moxham
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I like drawability in an abstract. Not just don’t mind it: actively prefer it, provided the draws aren’t dull and there aren’t too many of them. Of course there are cases that are decisive by nature – Hex, most famously – and I can’t imagine who would want to graft prosthetic draws onto those. But generally the idea of “honours even” appeals to the romantic in me, so much so that the absence of that possibility can feel like a gap in the game.

This has been my stance since forever. But recently there was a bubble of time in which, to my shame, I lost sight of it. I was musing on the fact that my new game Not Even Close was really just half a game – the Yin part, as it were - which it might be a humane act to reunite with its Yang (No Odd Neighbours), perhaps under the combined title NECNON. And it suddenly occurred to me that draws could be banished from the whole edifice by the addition of one small extra rule. This was a seductive thought and I promptly fell for it, announcing it as part of the rebranding.

There was a kind of logic (or so I told myself) to support the whim. I’d playtested NEC by pitting an AI against itself on a couple of different board sizes, and I wasn’t completely happy with the first/second player win ratio that seemed to be emerging. Nothing outrageous, but still – a six or seven percent imbalance in certain series. However, when I re-trialled with the anti-draw mechanism, that gap narrowed to two or three percent in each case. I couldn’t come up with any sort of explanation linking the improvement to the rule (with hindsight I ought to have read a warning into that), but since a link would be advantageous I more or less took it on trust. This is never [Edit: seldom - thank you, Christian] a wise move.

For a short while I was pleased with myself, and even went public with the rash assertion that NECNON was “flawless”. But comment by others variously denying or defending the merit of draws reignited my internal debate. It seemed on more careful reflection that a rule introduced to get rid of deadlock must ultimately be a blemish unless two things (maybe more, but two for starters) were both true. First, the goal set as a tie-breaker should flow naturally from mainstream strategy, rather than distracting from it; and second, it should be such that the player who achieved it could meaningfully be said to have deserved his victory.

I couldn’t persuade myself that either of these conditions was met by any mechanism I’d so far come up with, and the game was so stripped-back conceptually that I struggled to see where anything better might be hiding. Perhaps retention of the half-point bonus for last stone (which at least was clear-cut and completely transparent) might be justified at a pinch if it really did improve player balance along the way, but by now I had serious doubts about that too. My initial tests for draw-rate had been conducted over a few hundred games. At the time that seemed impressively thorough, but every subsequent test incorporating the tie-break (and there were two dozen of them) had used five thousand. Looked at dispassionately, there was good reason to wonder whether the supposed transformation of the stats was anything more than coincidental.

Only one honest course of action presented itself: re-test the original draws-and-all version over a much larger sample. So I did. I took both move protocols (NEC and NON), both types of grid (hexhex and square) and three well-spaced board sizes, and trialled each one of the twelve permutations ten thousand times. Even with five simultaneous series running 24/7 it took the better part of two weeks, and by the end my Mac had smoke-rings coming out of its ears, but it was worth it because the detailed results looked like this [click to magnify]:



The overall picture, whichever way you sliced it, was equally convincing:



So (to cut a long story not so very short) I’ve now reached my decision about NECNON. As far as the first player / second player issue goes, there isn’t much room for improvement in those figures. Truth is, if you catch yourself thinking of a 51.0%/49.0% series as a disappointing outlier, then you’re probably looking at a pretty well-balanced game! And as for the draw rates, I can’t see anything there to get particularly upset about either. I’m therefore officially removing the tie-break rule altogether, and I’d like to think – certainly I very much hope – that my doing so won’t necessarily turn anti-draw absolutists against a game I’m really rather proud of.

I’ll update the BGG NECNON page as soon as possible – hopefully later today. And then that will almost be that. I say “almost”, because I’d just like to investigate the viability of the natural third protocol: the one that has Player A NEC-ing while Player B NON-s. This, of course, means a good deal of additional testing, and I have a hunch it might not balance anyway. But more of that anon.
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christian freeling
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mocko wrote:

I like drawability in an abstract. Not just don’t mind it: actively prefer it, provided the draws aren’t dull and there aren’t too many of them.
After the last World Championship it occured to me that the draws were sometimes so thrilling because the defender's resources at times seemed miraculous. That's the game's merit as much as the player's. It's not easy to imagine a drawless game that might have a similar 'balance' although the ever present Hex goes a long way of course.

mocko wrote:
I couldn’t come up with any sort of explanation linking the improvement to the rule (with hindsight I ought to have read a warning into that), but since a link would be advantageous I more or less took it on trust. This is never a wise move.
Usually not, I agree, but 'never' is a bit harsh.

mocko wrote:
So (to cut a long story not so very short) I’ve now reached my decision about NECNON. As far as the first player / second player issue goes, there isn’t much room for improvement in those figures. Truth is, if you catch yourself thinking of a 51.0%/49.0% series as a disappointing outlier, then you’re probably looking at a pretty well-balanced game! And as for the draw rates, I can’t see anything there to get particularly upset about either. I’m therefore officially removing the tie-break rule altogether, and I’d like to think – certainly I very much hope – that my doing so won’t necessarily turn anti-draw absolutists against a game I’m really rather proud of.
The stats are impressive and I think you made the right decision.

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Craig Duncan
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Very impressive statistics, Richard! You're right to be proud of the game.

At the risk of seeming ungrateful... Do you have similar stats for hexhex5 and square 8? I am just curious, since those might be fairly common sizes to play on, what with Yavalath boards and Chess/Checkers boards floating around homes (the latter more so than the former, of course!).

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Richard Moxham
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cdunc123 wrote:
Very impressive statistics, Richard! You're right to be proud of the game.

At the risk of seeming ungrateful... Do you have similar stats for hexhex5 and square 8? I am just curious, since those might be fairly common sizes to play on, what with Yavalath boards and Chess/Checkers boards floating around homes (the latter more so than the former, of course!).

Thank you, Craig.

I started out intending to run those smaller board sizes, but I changed my mind in mid-stream. The reason was that I planned to limit myself to 'officially endorsing', as it were, just three sizes (though obviously people would be free to go with whatever they happened to possess the equipment for). And at a certain moment during the trialling process it occurred to me that the standard Go sizes of 9x9, 13x13 and 19x19, plus their nearest Hexhex equivalents, might be the most logical choices.

So as it happens, I do have results for Hexhex 5 (which IIRC were consistent with the others I've just published), but not for Square 8, because my change of tack occurred in between. But it's no sweat to run S8, so I'll do that and put out the results for both boards tomorrow morning.

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Craig Duncan
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mocko wrote:
cdunc123 wrote:
Very impressive statistics, Richard! You're right to be proud of the game.

At the risk of seeming ungrateful... Do you have similar stats for hexhex5 and square 8? I am just curious, since those might be fairly common sizes to play on, what with Yavalath boards and Chess/Checkers boards floating around homes (the latter more so than the former, of course!).

Thank you, Craig.

I started out intending to run those smaller board sizes, but I changed my mind in mid-stream. The reason was that I planned to limit myself to 'officially endorsing', as it were, just three sizes (though obviously people would be free to go with whatever they happened to possess the equipment for). And at a certain moment during the trialling process it occurred to me that the standard Go sizes of 9x9, 13x13 and 19x19, plus their nearest Hexhex equivalents, might be the most logical choices.

So as it happens, I do have results for Hexhex 5 (which IIRC were consistent with the others I've just published), but not for Square 8, because my change of tack occurred in between. But it's no sweat to run S8, so I'll do that and put out the results for both boards tomorrow morning.

Thank you, Richard.
 
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Richard Moxham
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Here are the stats I undertook to provide by this morning, for NECNON as played with either protocol on the Square 8 board (64 cells, obviously) and also on its nearest hexagonal equivalent HH 5 (61).

As with the previous results for larger boards, each of these four series consisted of 10,000 games, with an AI thinking time of 0.1 secs per move.

NEC on hexhex: Draws 1660 (= 1 in 6), Black 4148 (49.7%) Green 4192 (50.3%)
NON on hexhex: Draws 1250 (= 1 in 8), Black 4148 (50.3%) Green 4192 (49.7%)
NEC on square: Draws 1618 (= 1 in 6), Black 4111 (49.0%) Green 4271 (51.0%)
NON on square: Draws 864 (= 1 in 12), Black 4521 (49.5%) Green 4615 (50.5%)

So as you see, the balance is as good as ever here, and the draw rate still pretty acceptable, assuming you can tolerate draws at all.

I would have expected to be adding the caveat that these boards are probably too small for really serious play, but it turns out that Russ hasn't (or anyway hadn't) beaten Stephen's Ai even once on Hexhex 5, and I certainly haven't either . Anyone else have experience to report?

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Craig Duncan
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mocko wrote:

Here are the stats I undertook to provide by this morning, for NECNON as played with either protocol on the Square 8 board (64 cells, obviously) and also on its nearest hexagonal equivalent HH 5 (61).

As with the previous results for larger boards, each of these four series consisted of 10,000 games, with an AI thinking time of 0.1 secs per move.

NEC on hexhex: Draws 1660 (= 1 in 6), Black 4148 (49.7%) Green 4192 (50.3%)
NON on hexhex: Draws 1250 (= 1 in 8), Black 4148 (50.3%) Green 4192 (49.7%)
NEC on square: Draws 1618 (= 1 in 6), Black 4111 (49.0%) Green 4271 (51.0%)
NON on square: Draws 864 (= 1 in 12), Black 4521 (49.5%) Green 4615 (50.5%)

So as you see, the balance is as good as ever here, and the draw rate still pretty acceptable, assuming you can tolerate draws at all.

I would have expected to be adding the caveat that these boards are probably too small for really serious play, but it turns out that Russ hasn't (or anyway hadn't) beaten Stephen's Ai even once on Hexhex 5, and I certainly haven't either . Anyone else have experience to report?

Thank you for the additional data, Richard. Good know about the balance.

Interesting to me that draws are less common in NON than NEC. I guess life is easier in NEC.

Draw rates aside, do you find that you have a preference between NON and NEC, for whatever reason? And for either game, do you have a preference between hex board vs square board?
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Richard Moxham
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cdunc123 wrote:
Thank you for the additional data, Richard. Good know about the balance.

Interesting to me that draws are less common in NON than NEC. I guess life is easier in NEC.

Draw rates aside, do you find that you have a preference between NON and NEC, for whatever reason? And for either game, do you have a preference between hex board vs square board?
Yes, the balance is good, isn't it? And it's nice to see that HH5 and SQ8 are both properly playable.

But I think that's pretty much as low as you can really go. Just out of curiosity I ran a single 10k series on what is effectively the next size down: HH4 (37 cells)/ SQ6 (36). (I used the square board, simply because I happened to be working with that at the time.) Anyway, the balance maintained its usual uncanny evenness (Black 51.3%, Green 48.7%), but the problem lay in the other important division - i.e. draws v decisives. In its way this was rather uncanny too (49.4% v 50.6%), but of course that meant a draw every second game, which I think everyone would consider a bit much.

In general, the constancy of statistical patterns is one of the most noticeable features of the game. You mention the NEC/NON comparison, and indeed we can see pretty clear recurrence there, with NEC draws as frequent or only very slightly rarer on squares as compared with hexes, beside which NON draws are less frequent than NEC by a modest amount on hexes but then much rarer on the squares. And this appears to hold for all four board sizes for which we have complete figures. I haven't got anywhere near the bottom of the draw disparity, but my current hunch is that it may be related to the degree to which the respective protocols conduce to the formation of air holes. There's also the issue of the extent to which human v human play would replicate the AI v AI draw rates.

As for my personal preferences, I can't say I've mastered play in either case sufficiently to be entitled to one. I always like odd numbers more than even ones, but only in the same way as I support Oxford in the Boat Race on the grounds that dark blue is obviously more beautiful than light

[Edit: Oh yes. And I guess the hex board appeals to me more in this particular case (hexes for placement, squares for movement), but I'd also be interested to see what someone familiar with Go thought of the comparison there.)


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Craig Duncan
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mocko wrote:
[Edit: Oh yes. And I guess the hex board appeals to me more in this particular case (hexes for placement, squares for movement), but I'd also be interested to see what someone familiar with Go thought of the comparison there.)
I haven't tried square grid NECNON yet. My guess is that I will prefer the hex board, largely for reasons of clarity. On a square grid one must ignore diagonal neighbors when counting for evens or odds among orthogonal neighbors; on a hex grid all neighbors count. I also suspect that since hex boards allow for three forbidden counts (2,4,6 in NEC; 1,3,5 in NON) compared to two forbidden counts on a square grid (2,4; 1,3), hex grids may allow for richer tactics.

That said, it appears that grids have the virtue of leading to fewer draws on average. That's surely because there are only four liberties to close off, as opposed to six on the hex board, and thus it is easier to smother one's opponent's stones. But I'm not sure that virtue is enough to outmatch (what I suspect to be) the hex board's other virtues.
 
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Richard Moxham
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cdunc123 wrote:
mocko wrote:
[Edit: Oh yes. And I guess the hex board appeals to me more in this particular case (hexes for placement, squares for movement), but I'd also be interested to see what someone familiar with Go thought of the comparison there.)
I haven't tried square grid NECNON yet. My guess is that I will prefer the hex board, largely for reasons of clarity. On a square grid one must ignore diagonal neighbors when counting for evens or odds among orthogonal neighbors; on a hex grid all neighbors count. I also suspect that since hex boards allow for three forbidden counts (2,4,6 in NEC; 1,3,5 in NON) compared to two forbidden counts on a square grid (2,4; 1,3), hex grids may allow for richer tactics.

That said, it appears that grids have the virtue of leading to fewer draws on average. That's surely because there are only four liberties to close off, as opposed to six on the hex board, and thus it is easier to smother one's opponent's stones. But I'm not sure that virtue is enough to outmatch (what I suspect to be) the hex board's other virtues.
You may very well be right on all counts, Craig. All of this will have to be explored. I must admit that I myself think of hexhex as the core game (it certainly arrived first) - though I'm also very pleased with the fact that NN transfers seamlessly to the square board (and for all I know to the tessellation of octagons and squares or what not else). Incidentally, for reasons you touch upon, I'd ultimately prefer to see the square version played on the intersections, like Go. I think we know that that improves clarity.
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Craig Duncan
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mocko wrote:
Incidentally, for reasons you touch upon, I'd ultimately prefer to see the square version played on the intersections, like Go. I think we know that that improves clarity.
Yes, I agree. Playing on intersections is sure the best way to play on a square grid.
 
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Richard Moxham
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Quote:
I haven't tried square grid NECNON yet. My guess is that I will prefer the hex board, largely for reasons of clarity. On a square grid one must ignore diagonal neighbors when counting for evens or odds among orthogonal neighbors; on a hex grid all neighbors count. I also suspect that since hex boards allow for three forbidden counts (2,4,6 in NEC; 1,3,5 in NON) compared to two forbidden counts on a square grid (2,4; 1,3), hex grids may allow for richer tactics.

Craig’s intuition regarding comparative tactical richness may be accurate, though it’s not obvious to me that this need be so. Might turn out to be just a different style of war. Be that as it may, his remarks rekindled my curiosity about the other option implicit in the square grid: i.e., eightfold adjacency. So I ran viability tests for that on 19x19, 13x13 and 9x9. And then (for reasons I’ll explain below) the thought occurred that I should just check out 18x18, 12x12 and 8x8 while the urge was upon me. The results are shown in this table, with the fourfold connectivity outcomes (already published earlier in this thread) alongside for comparison.



So now we know that combining orthogonal and diagonal on squares works just as well as all the other permutations tested to date. Twenty-four series altogether – just short of a quarter of a million games – and not one where the percentage Black/Green disparity exceeds 51.0/49.0 in either direction. (If this testifies, as it surely does, to the merits of the NECNON protocols, it’s also a glowing advertisement for 122* turn order.) The draw rates, too, are fine by me – even the ‘low’ of 75% decisiveness for 8x8, which I think is in any case a board likely to see most use by newcomers on their way through to greater mastery.

If a physical set of NECNON were ever to be published, I’d like to see a double-sided board with the hexes on one side and the squares on the other, smaller board sizes being differentiated within the larger by light but easily perceptible gradations of shading. On the square grid, from the point of view of visual clarity, fourfold adjacency would naturally lend itself to play on the intersections, and eightfold to the cells – and it was realising this that led me to re-test the latter at 18, 12 and 8.

To sum up, NECNON seems to me very close to attaining its final form. The only piece of research remaining to be conducted is the viability test for NEC versus NON within a single game. I’d like to run this while the project is still fresh and while I’m still accustomed to having my computer tied up for whole weeks at a time. But, as I’ve explained previously, I can’t go any further without another injection of technical assistance from Stephen Tavener, and can’t ask him to oblige yet again without some sort of indication that the AbStrat community would consider it to be of interest. So my thanks are due to those contributors who have already helped move NECNON from Page 3 to Page 1 on the AiAi priority list, and if there are half a dozen or so others prepared to lend a thumb in the cause of science I’d be extremely grateful.

(Afterthought: I should perhaps have made clear that eightfold adjacency governs not merely the placement protocol but also the connectivity of groups, so it’s a very different proposition indeed from its fourfold sibling.)

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Stephen Tavener
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Richard, you should use the time while you wait to run with longer thinking times. Possibly MUCH longer times.

If I recall correctly, you are running with 0.1s/move (which is not very long), on reasonably old hardware. As the board size increases not only does the branching factor at each move increase, but also the length of each move; so the AI gets _much_ weaker.

Imagine a game with free placement of pieces like NECNON where there are 5 cells. That makes for 5! possible random playouts = 120. Now imagine the same game with 10 cells. There are now 10! possible random playouts, which is 3,628,800 possibilities. I'm not saying that Ai Ai gets 30,240 times weaker when you double the board size from 5 to 10, but I AM saying that Ai Ai can see a much smaller portion of the game tree, so the results become much less reliable for the same thinking time. How many iterations/move are you seeing on the larger boards?

Next note that balance is an illusion. A combinatorial game like NECNON is solvable (but not solved) so a perfect player would always reach the same outcome, every single game. Tat the playouts are so balanced, tells you that Ai Ai isn't close to solving the game - but that doesn't mean a human would play in the same way, or see the same results.

Finally, I should also point out that a coin toss will give almost 50% wins to each player with no possibility for draws, yet not very interesting to play. A reasonable level of balance may be a requirement for a good game, but is not in itself an indicator.

In summary, while I think Ai Ai can be a valuable sanity check when testing a game, and may give a nod and a wink in the right direction, it should not replace testing against humans.
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Richard Moxham
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Stephen, thank you very much for these reminders, to which be assured I give serious heed.

As far as the thinking time’s concerned, I’m naturally all too conscious that 0.1 secs isn’t long, but it’s a bit of a trade-off, isn’t it? You yourself have said on past occasions that anything less than 1000 plays is a dubious sample, and I’ve been running parallel series of only 2000 – which even then on the large boards take over 24 hours non-stop. I have run (and in some cases played) individual much longer games, but I don’t have unlimited time (or indeed computer time) to devote to this, and the question is how many games at, say, 5 secs per move would constitute anything like significance, given that a series of even 1000 on a 13x13 board would then last getting on for a month?

As for balance, obviously it’s an illusion in the sense that, since every variant line is completely determined, the existence of even one unanswerable opening move makes the game a first-player win, and it’s otherwise a draw unless unanswerable second-player cover exists for every single opener. Looked at that way, a game is of course wholly win, lose or draw and the concept of distributed advantage is meaningless. All this I get. But I would argue that there’s another sense – admittedly more dilute but still meaningful in the mortal realm – which could be defined as a game’s propensity for the even distribution of error. At any rate, my suspicion is that NECNON’s a draw in absolute terms, but that Black’s and Green’s win records will be very similar (and will far outstrip the draw) for a long time to come.

So all in all, I wouldn’t claim the stats from all this trialling to be proof of anything, but you can’t do more than pass the tests you take and I doubt that every game would have come out of these as satisfactorily. We’ll just have to see, when (if?) it finds its way onto a forum where a lot of people play a lot of other people, how good a game the punters judge it to be.

Anyway, grateful for the constructive comment, as I said above (’cept I’ll thank you not to diss my not-a-day-over-two-year-old MacBook Pro ).
 
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Stephen Tavener
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mocko wrote:
As far as the thinking time’s concerned, I’m naturally all too conscious that 0.1 secs isn’t long, but it’s a bit of a trade-off, isn’t it?
But do you know what you're trading off? My concern is that on larger board sizes, Ai AI will be behaving more or less randomly - but without knowing how many iterations is is getting for each move, there is no way to tell.

Quote:
You yourself have said on past occasions that anything less than 1000 plays is a dubious sample
Correct. 1000 gives +/- about 3% at the 95 confidence level.

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How many games at, say, 5 secs per move would constitute anything like significance, given that a series of even 1000 on a 13x13 board would then last getting on for a month?
The time is irrelevant to the accuracy.

On the subject of tests, make sure you test mirror play on the square boards with even sides.

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I’ll thank you not to diss my not-a-day-over-two-year-old MacBook Pro )
I'll hold out for the its/sec rating before dissing it.
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mrraow wrote:
On the subject of tests, make sure you test mirror play on the square boards with even sides.

Is it really necessary to consider this when leapfrog turn-order is in force? Even supposing the placement protocols are susceptible, which doesn't seem completely obvious to me, wouldn't both players have to be absolutely determined from start to finish to engineer the draw? Or am I missing something?

[Edit: and how would one test for it?]
 
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Brian Svoboda
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This suggestion probably won't be helpful unless you already have Linux system administrator skills, but you could rent out a CPU optimized cloud-computer instance on DigitalOcean (DO). This would free-up your laptop and also be faster depending on which kind of instance you rent. They're much easier to use than Amazon AWS, and DO has superb documentation. I used the smallest DO "droplet" as a custom VPN for a while, it was actually pretty easy! Checking their pricing page shows 8 virtual-CPU cores for $0.238/hr up to a maximum of $160/month.

If you're into computers, AMD also makes very good and cheap options if you are just interested in fully-parallel tasks. The AMD Ryzen 2600 has 6 physical cores with a 3.4 GHz clock speed at $160 dollars! Wow.

 
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Stephen Tavener
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Ah, yes - I forgot it was a 12* game. You're safe.
 
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Richard Moxham
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mrraow wrote:
mocko wrote:
How many games at, say, 5 secs per move would constitute anything like significance, given that a series of even 1000 on a 13x13 board would then last getting on for a month?
The time is irrelevant to the accuracy.
Sorry, Stephen - I'm probably being dense, but would you mind unpacking this answer a little?

 
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Stephen Tavener
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mocko wrote:
mrraow wrote:
mocko wrote:
How many games at, say, 5 secs per move would constitute anything like significance, given that a series of even 1000 on a 13x13 board would then last getting on for a month?
The time is irrelevant to the accuracy.
Sorry, Stephen - I'm probably being dense, but would you mind unpacking this answer a little?
If you have a sequence of result; { win, loss, draw, draw, win, win, etc. } it will give the same win percentage and variance regardless of the thinking time; in other words, for +/- 3% at the 95% confidence level you need around 1,000 games regardless of thinking time.

* It's not that simple of course; { draw, draw, draw, ... } will give much tighter confidence bounds than { win, draw, win, draw, ... } even though the mean is the same; more consistency => tighter bounds.
 
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